Contraction (operator theory)

In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T || ≤ 1. This notion is a special case of the concept of a contraction mapping, but every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.

If T is a contraction acting on a Hilbert space ${\displaystyle {\mathcal {H}}}$, the following basic objects associated with T can be defined.

The defect operators of T are the operators D_T = (1 − T*T)^½ and D_T* = (1 − TT*)^½. The square root is the positive semidefinite one given by the spectral theorem. The defect spaces ${\displaystyle {\mathcal {D}}_{T}}$ and ${\displaystyle {\mathcal {D}}_{T*}}$ are the closure of the ranges Ran(D_T) and Ran(D_T*) respectively. The positive operator D_T induces an inner product on ${\displaystyle {\mathcal {H}}}$. The inner product space can be identified naturally with Ran(D_T). A similar statement holds for ${\displaystyle {\mathcal {D}}_{T*}}$.

The defect indices of T are the pair

The defect operators and the defect indices are a measure of the non-unitarity of T.

A contraction T on a Hilbert space can be canonically decomposed into an orthogonal direct sum

where U is a unitary operator and Γ is completely non-unitary in the sense that it has no non-zero reducing subspaces on which its restriction is unitary. If U = 0, T is said to be a completely non-unitary contraction. A special case of this decomposition is the Wold decomposition for an isometry, where Γ is a proper isometry.

Contractions on Hilbert spaces can be viewed as the operator analogs of cos θ and are called operator angles in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.